We say that a natural number $ n\ge 4 $ is [i]unusual[/i] if, for any $ n\times n $ array of real numbers, the sum of the numbers from any $ 3\times 3 $ compact subarray is negative, and the sum of the numbers from any $ 4\times 4 $ compact subarray is positive. Find all unusual numbers.

Compute $1+2\cdot3^4$. [i]Proposed by Evan Chen[/i]

Prove that there are infinitely many positive integers $ n$ such that $ n^{2} \plus{} 1$ has a prime divisor greater than $ 2n \plus{} \sqrt {2n}$. [i]Author: Kestutis Cesnavicius, Lithuania[/i]

A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed "9" and you pressed [+1], it would display "10." If you then pressed [x2], it would display "20." Starting with the display "1," what is the fewest number of keystrokes you would need to reach "200"? $ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$

Find the number of pairs $(n,C)$ of positive integers such that $C\leq 100$ and $n^2+n+C$ is a perfect square.

If one minus the reciprocal of $(1-x)$ equals the reciprocal of $(1-x)$, then $x$ equals $\textbf{(A) }-2\qquad\textbf{(B) }-1\qquad\textbf{(C) }1/2\qquad\textbf{(D) }2\qquad \textbf{(E) }3$

Let $ABC$ be an acute triangle with orthocenter $H$, and let $BE$ and $CF$ be the altitudes of the triangle. Choose two points $P$ and $Q$ on rays $BH$ and $CH$ respectively, such that: $\bullet$ $PQ$ is parallel to $BC$; $\bullet$ The quadrilateral $APHQ$ is cyclic. Suppose the circumcircles of triangles $APF$ and $AQE$ meet again at $X\neq A$. Prove that $AX$ is parallel to $BC$. [i]Proposed by Ivan Chan Kai Chin[/i]

Let $ K$ be a compact topological group, and let $ F$ be a set of continuous functions defined on $ K$ that has cardinality greater that continuum. Prove that there exist $ x_0 \in K$ and $ f \not\equal{}g \in F$ such that \[ f(x_0)\equal{}g(x_0)\equal{}\max_{x\in K}f(x)\equal{}\max_{x \in K}g(x).\] [i]I. Juhasz[/i]

Triangle $ABC$ is right angled at $A$. The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$, determine $AC^2$.

Let $m, n, k$ be positive integers with $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $1, 2, ... , n$ can be divided into $k$ groups in such a way that the sum of the numbers in each group equals $m$.

[b]p1.[/b] Len's Spanish class has four tests in the first term. Len scores $72$, $81$, and $78$ on the first three tests. If Len wants to have an 80 average for the term, what is the minimum score he needs on the last test? [b]p2.[/b] In $1824$, the Electoral College had $261$ members. Andrew Jackson won $99$ Electoral College votes and John Quincy Adams won $84$ votes. A plurality occurs when no candidate has more than $50\%$ of the votes. Should a plurality occur, the vote goes to the House of Representatives to break the tie. How many more votes would Jackson have needed so that a plurality would not have occurred? [b]p3.[/b] $\frac12 + \frac16 + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}= 1 - \frac{1}{n}$. Find $n$. [b]p4.[/b] How many ways are there to sit Samuel, Esun, Johnny, and Prat in a row of $4$ chairs if Prat and Johnny refuse to sit on an end? [b]p5.[/b] Find an ordered quadruple $(w, x, y, z)$ that satisfies the following: $$3^w + 3^x + 3^y = 3^z$$ where $w + x + y + z = 2017$. [b]p6.[/b] In rectangle $ABCD$, $E$ is the midpoint of $CD$. If $AB = 6$ inches and $AE = 6$ inches, what is the length of $AC$? [b]p7.[/b] Call an integer interesting if the integer is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$, so $27$ is interesting. How many $2$-digit interesting integers are there? [b]p8.[/b] Let $a\#b = \frac{a^3-b^3}{a-b}$ . If $a, b, c$ are the roots of the polynomial $x^3 + 2x^2 + 3x + 4$, what is the value of $a\#b + b\#c + c\#a$? [b]p9.[/b] Akshay and Gowri are examining a strange chessboard. Suppose $3$ distinct rooks are placed into the following chessboard. Find the number of ways that one can place these rooks so that they don't attack each other. Note that two rooks are considered attacking each other if they are in the same row or the same column. [img]https://cdn.artofproblemsolving.com/attachments/f/1/70f7d68c44a7a69eb13ce12291c0600d11027c.png[/img] [b]p10.[/b] The Earth is a very large sphere. Richard and Allen have a large spherical model of Earth, and they would like to (for some strange reason) cut the sphere up with planar cuts. If each cut intersects the sphere, and Allen holds the sphere together so it does not fall apart after each cut, what is the maximum number of pieces the sphere can be cut into after $6$ cuts? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Initially, a spinner points west. Chenille moves it clockwise $ 2 \dfrac{1}{4}$ revolutions and then counterclockwise $ 3 \dfrac{3}{4}$ revolutions. In what direction does the spinner point after the two moves? [asy]size(100); draw(circle((0,0),1),linewidth(1)); draw((0,0.75)--(0,1.25),linewidth(1)); draw((0,-0.75)--(0,-1.25),linewidth(1)); draw((0.75,0)--(1.25,0),linewidth(1)); draw((-0.75,0)--(-1.25,0),linewidth(1)); label("$N$",(0,1.25), N); label("$W$",(-1.25,0), W); label("$E$",(1.25,0), E); label("$S$",(0,-1.25), S); draw((0,0)--(-0.5,0),EndArrow);[/asy] $ \textbf{(A)}\ \text{north} \qquad \textbf{(B)}\ \text{east} \qquad \textbf{(C)}\ \text{south} \qquad \textbf{(D)}\ \text{west} \qquad \textbf{(E)}\ \text{northwest}$

Prove that a $46$-element set of integers contains two distinct doubletons $\{u, v\}$ and $\{x,y\}$ such that $u + v \equiv x + y$ (mod $2016$).

Circumcircle of $\triangle ABC$ is $\odot O$, incentre of $\triangle ABC$ is $I$. $\angle B=60^{\circ}.\angle A<\angle C$. Bisector of outer angle $\angle A$ intersects $\odot O$ at $E$. Prove: [b](a)[/b] $IO=AE$. [b](b)[/b] The radius of $\odot O$ is $R$, then $2R<IO+IA+IC<(1+\sqrt3)R$.

Let quadrilateral $A_1A_2A_3A_4$ be not cyclic and haves edges not parallel to each other. Denote $B_i$ as the intersection of the tangent line at $A_i$ with respect to circle $A_{i-1}A_iA_{i+1}$ and the $A_{i+2}$-symmedian with respect to triangle $A_{i+1}A_{i+2}A_{i+3}$ and $C_i$ as the intersection of lines $A_iA_{i+1}$ and $B_iB_{i+1}$, where all indexes taken cyclically. Prove that $C_1$, $C_2$, $C_3$, and $C_4$ are collinear.

Do either $(1)$ or $(2)$: $(1)$ Prove that $\int_{1}^{k} [x] f'(x) dx = [k] f(k) - \sum_{1}{[k]} f(n),$ where $k > 1,$ and $[z]$ denotes the greatest integer $\leq z.$ Find a similar expression for: $\int_{1}^{k} [x^2] f'(x) dx.$ $(2)$ A particle moves freely in a straight line except for a resistive force proportional to its speed. Its speed falls from $1,000 \dfrac{ft}{s}$ to $900 \dfrac{ft}{s}$ over $1200 ft.$ Find the time taken to the nearest $0.01 s.$ [No calculators or log tables allowed!]

Let $ABCD$ be a convex quadrilateral, with $\measuredangle ABC > 90^\circ$, $\measuredangle CDA > 90^\circ$ and $\measuredangle DAB = \measuredangle BCD$. Let $E$, $F$ and $G$ be the reflections of $A$ with respect to the lines $BC$, $CD$ and $DB$. Finally, let the line $BD$ meet the line segment $AE$ at a point $K$, and the line segment $AF$ at a point $L$. Prove that the circumcircles of the triangles $BEK$ and $DFL$ are tangent to each other at $G$.

In the figure, $BC$ is a diameter of the circle, where $BC=\sqrt{257}$, $BD=1$ and $DA=12$. Find the length of $EC$ and hence find the length of the altitude from $A$ to $BC$. [asy] import cse5; size(200); pair O=(2, 0), B=(0, 0), C=(4, 0), A=(1, 3), D, E; D=MP("D",D(IP(D(CP(O,B)),D(MP("A",D(A),N)--MP("B",D(B),W)))),NW); E=MP("E",D(IP(CP(O,B),D(MP("C",D(C),NE)--A),1)),NE); D(B--C); [/asy]

For any positive integer $n$, let $ \left< n \right> $ denote the closest integer to $ \sqrt {n} $. Evaluate: \[ \displaystyle\sum_{n=1}^{\infty} \dfrac {2^{\left< n \right>} + 2^{- \left< n \right>}}{2^n} \]

At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, how many campers are in exactly two classes? $\text{(A) }10\qquad\text{(B) }11\qquad\text{(C) }12\qquad\text{(D) }13\qquad\text{(E) }14$

If $a \diamondsuit b = \vert a - b \vert \cdot \vert b - a \vert$ then find the value of $1 \diamondsuit (2 \diamondsuit (3 \diamondsuit (4 \diamondsuit 5)))$. [i]Proposed by Muztaba Syed[/i] [hide=Solution] [i]Solution.[/i] $\boxed{9}$ $a\diamondsuit b = (a-b)^2$. This gives us an answer of $\boxed{9}$. [/hide]

The extensions of sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ intersect at $K$. It is known that $AD = BC$. Let $M$ and $N$ be the midpoints of sides $AB$ and $CD$. Prove that the triangle $MNK$ is obtuse. [i](5 points)[/i]

Show that there is a set of $2002$ consecutive positive integers containing exactly $150$ primes. (You may use the fact that there are $168$ primes less than $1000$)

In $\triangle ABC$, $AB<AC$. $D$ and $P$ are the feet of the internal and external angle bisectors of $\angle BAC$, respectively. $M$ is the midpoint of segment $BC$, and $\omega$ is the circumcircle of $\triangle APD$. Suppose $Q$ is on the minor arc $AD$ of $\omega$ such that $MQ$ is tangent to $\omega$. $QB$ meets $\omega$ again at $R$, and the line through $R$ perpendicular to $BC$ meets $PQ$ at $S$. Prove $SD$ is tangent to the circumcircle of $\triangle QDM$. [i]Proposed by Ray Li[/i]

If, instead, the graph is a graph of ACCELERATION vs. TIME and the squirrel starts from rest, then the squirrel has the greatest speed at what time(s) or during what time interval? (A) at B (B) at C (C) at D (D) at both B and D (E) From C to D